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Properties

Label 20.8.40198718926...0000.1
Degree $20$
Signature $[8, 6]$
Discriminant $2^{28}\cdot 3^{16}\cdot 5^{16}\cdot 691^{4}$
Root discriminant $85.16$
Ramified primes $2, 3, 5, 691$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 20T1025

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![12960000, 0, 25920000, 0, -2304000, 0, -5256000, 0, 410000, 0, 381000, 0, -71880, 0, 3190, 0, 75, 0, -15, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^20 - 15*x^18 + 75*x^16 + 3190*x^14 - 71880*x^12 + 381000*x^10 + 410000*x^8 - 5256000*x^6 - 2304000*x^4 + 25920000*x^2 + 12960000)
 
gp: K = bnfinit(x^20 - 15*x^18 + 75*x^16 + 3190*x^14 - 71880*x^12 + 381000*x^10 + 410000*x^8 - 5256000*x^6 - 2304000*x^4 + 25920000*x^2 + 12960000, 1)
 

Normalized defining polynomial

\( x^{20} - 15 x^{18} + 75 x^{16} + 3190 x^{14} - 71880 x^{12} + 381000 x^{10} + 410000 x^{8} - 5256000 x^{6} - 2304000 x^{4} + 25920000 x^{2} + 12960000 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $20$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[8, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(401987189260024679669760000000000000000=2^{28}\cdot 3^{16}\cdot 5^{16}\cdot 691^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $85.16$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $2, 3, 5, 691$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{20} a^{8} - \frac{1}{4} a^{6} + \frac{1}{4} a^{4}$, $\frac{1}{20} a^{9} - \frac{1}{4} a^{7} + \frac{1}{4} a^{5}$, $\frac{1}{40} a^{10} - \frac{1}{40} a^{8} + \frac{1}{8} a^{6} - \frac{1}{2} a^{4}$, $\frac{1}{40} a^{11} - \frac{1}{40} a^{9} + \frac{1}{8} a^{7} - \frac{1}{2} a^{5}$, $\frac{1}{400} a^{12} - \frac{1}{80} a^{10} + \frac{1}{80} a^{8} - \frac{3}{20} a^{6} + \frac{1}{20} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{400} a^{13} - \frac{1}{80} a^{11} + \frac{1}{80} a^{9} - \frac{3}{20} a^{7} + \frac{1}{20} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{2400} a^{14} - \frac{1}{800} a^{12} + \frac{1}{160} a^{10} + \frac{1}{240} a^{8} - \frac{2}{5} a^{4} + \frac{1}{3} a^{2}$, $\frac{1}{2400} a^{15} - \frac{1}{800} a^{13} + \frac{1}{160} a^{11} + \frac{1}{240} a^{9} - \frac{2}{5} a^{5} + \frac{1}{3} a^{3}$, $\frac{1}{72000} a^{16} - \frac{1}{4800} a^{14} + \frac{1}{960} a^{12} - \frac{41}{7200} a^{10} + \frac{1}{600} a^{8} - \frac{5}{24} a^{6} - \frac{1}{18} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{72000} a^{17} - \frac{1}{4800} a^{15} + \frac{1}{960} a^{13} - \frac{41}{7200} a^{11} + \frac{1}{600} a^{9} - \frac{5}{24} a^{7} - \frac{1}{18} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{401802526310784942096000} a^{18} - \frac{237712421267291083}{44644725145642771344000} a^{16} - \frac{154683265761684007}{2060525775952743292800} a^{14} - \frac{47159504657307069917}{40180252631078494209600} a^{12} - \frac{11488868459830288757}{2232236257282138567200} a^{10} - \frac{5250578887114609013}{837088596480801962700} a^{8} + \frac{15469110837210909647}{251126578944240588810} a^{6} - \frac{26287373183800822049}{83708859648080196270} a^{4} - \frac{168801736723422857}{5580590643205346418} a^{2} - \frac{203640968129745415}{930098440534224403}$, $\frac{1}{401802526310784942096000} a^{19} - \frac{237712421267291083}{44644725145642771344000} a^{17} - \frac{154683265761684007}{2060525775952743292800} a^{15} - \frac{47159504657307069917}{40180252631078494209600} a^{13} - \frac{11488868459830288757}{2232236257282138567200} a^{11} - \frac{5250578887114609013}{837088596480801962700} a^{9} + \frac{15469110837210909647}{251126578944240588810} a^{7} - \frac{26287373183800822049}{83708859648080196270} a^{5} - \frac{168801736723422857}{5580590643205346418} a^{3} - \frac{203640968129745415}{930098440534224403} a$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $13$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 778604444074 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

20T1025:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 7372800
The 216 conjugacy class representatives for t20n1025 are not computed
Character table for t20n1025 is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), 10.6.5438807015625.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 20 sibling: data not computed
Degree 40 siblings: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R R R ${\href{/LocalNumberField/7.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ $16{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.8.0.1}{8} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/29.5.0.1}{5} }^{4}$ ${\href{/LocalNumberField/31.10.0.1}{10} }{,}\,{\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{4}$ $16{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ $16{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }$ $16{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{2}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
2Data not computed
$3$3.8.4.2$x^{8} - 27 x^{2} + 162$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$
3.12.12.6$x^{12} + 24 x^{11} - 3 x^{10} + 81 x^{9} - 18 x^{8} + 54 x^{7} + 108 x^{5} - 54 x^{4} - 27 x^{3} - 81 x - 81$$3$$4$$12$12T39$[3/2, 3/2]_{2}^{4}$
$5$5.8.7.1$x^{8} - 5$$8$$1$$7$$C_8:C_2$$[\ ]_{8}^{2}$
5.12.9.1$x^{12} - 10 x^{8} - 375 x^{4} - 2000$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$
691Data not computed